73. Find the area between the curves y=ln(x) and y=ln(2x) from...

Question

73. Find the area between the curves y=ln(x) and y=ln(2x) from x=1 to x=5.

Accepted Answer

Welcome back everyone. Find the area between the curves Y equals ln 3 X and Y equals Lnx from x equals 22 x equals 6. A 2 LN 3 square units, B4 LN 2 square units, C 2 LN 2 square units, and D 4 LN 3 square units. For this problem, let's remember the area formula area is equal to the integral from x1 up to x2 of the difference between f of x, which is the upper curve, and g of x, which is the lower curve. We're going to integrate with respect to x. Now how do we determine which one is the upper curve? We're going to start with the first function y equals ln 3x, and using the properties of logarithms we can express it as LN 3 plus LN x. And the other curve is a line of X. Now LN of 3 is a positive number, so notice that we're adding a positive number to LN of X, meaning LN of 3 plus LN X is going to be greater than LN of X within the domain, right? X is greater than 0. So we can now show that the upper curve F of X is L line of 3 X. And the lower curve is going to be LN of X. So now the area is going to be the integral from 2 to 6. This is what the problem suggests. And we're going to integrate the difference between LN of 3X and LNX. Now let's apply the properties of logarithms once again. We have integral from 2 to 6, and we can replace the difference. Using the quotient rule, right, that would be ln of 3 x divided by x d x. We can now cancel out the Xs and we're basically integrating LN of 3 DX. LN of 3 is a constant. So we get a line of 3 integral from 2 to 6 d x. The integral of d x is simply x, so we get xln of 3. Evaluate that from 2 up to 6. And now we can apply the fundamental theorem of calculus part 2. Eon of 3 is a constant, so we just want to find the difference between 6 and 2, right? Because we have X only. So 6 minus 2, this is what we have in parent multiplied by eon of 3. This gives us 41 of 3. So the final answer corresponds to the answer to the 4 elon of 3 square units. Thank you for watching.

73. Find the area between the curves y=ln(x) and y=ln(2x) from x=1 to x=5.

Watch next